Tuesday, March 13, 2007

070312_HoneyComb (re-scripting)

TILINGA plane-filling arrangement of plane figures or its generalization to higher dimensions. Formally, a tiling is a collection of disjoint open sets, the closures of which cover the plane. Given a single tile, the so-called first corona is the set of all tiles that have a common boundary point with the tile (including the original tile itself).

Wang's conjecture (1961) stated that if a set of tiles tiled the plane, then they could always be arranged to do so periodically. A periodic tiling of the plane by polygons or space by polyhedra is called a tessellation. The conjecture was refuted in 1966 when R. Berger showed that an aperiodic set of 20426 tiles exists. By 1971, R. Robinson had reduced the number to six and, in 1974, R. Penrose discovered an aperiodic set (when color-matching rules are included) of two tiles: the so-called Penrose tiles. It is not known if there is a single aperiodic tile.(ie mathworld.com)HONEY COMBAs part of a re-visiting codes series -most often following students requirments- I went back to one of my very first rhinoscript code generating Honeycomb cells onto a host nurbs surface and rewrote it... mainly shorter, for sure much faster and somehow within a concern of a certain "elegance" -to quote the latest trend of fitness criteria accoding to Ali Rahim's latest issue of AD magasine!- as an exercice within the exercice...

Tips and tricks:The generation of the honeycomb cells within rhinoscript is now defined by one unique conditional statment using many booleans operations.If (i>1 And j>1) And ( ( ((i-1)Mod 2) And (((j-2)Mod 4)= 0) ) Or ( (((i-1)Mod 2)=0) And ((j Mod 4)=0) ) Or ( (((i-1)Mod 2)=0) And (((j-1)Mod 4)=0) ) Or ( (((i-1)Mod 2)) And (((j+1)Mod 4)=0) ) )HONEYCOMB GEOMETRY: (ie wikipedia.org)The axes of honeycomb cells are always quasi-horizontal, and the non-angled rows of honeycomb cells are always horizontally (not vertically) aligned. Thus, each cell has two vertical walls, with "floors" and "ceilings" composed of two angled walls. The cells slope slightly upwards, between 9 and 14 degrees, towards the open ends.

There are two possible explanations for the reason that honeycomb is composed of hexagons, rather than any other shape. One, given by Jan Brożek, is that the hexagon tiles the plane with minimal surface area. Thus a hexagonal structure uses the least material to create a lattice of cells with a given volume. Another, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles. In support of this he notes that queen cells, which are constructed singly, are irregular and lumpy with no apparent attempt at efficiency.

It is likely that the honey bee constructs the honeycomb based on instinct, and the prevailing theory of biology is that the appearance of such efficient shapes in nature is a result of natural selection.

The closed ends of the honeycomb cells are also an example of geometric efficiency, albeit three-dimensional and little-noticed. The ends are trihedral (i.e., composed of three planes) pyramidal in shape, with the dihedral angles of all adjacent surfaces measuring 120°, the angle that minimizes surface area for a given volume. (The angle formed by the edges at the pyramidal apex is approximately 109° 28' 16" (= 180° - arccos(1/3)).)

The shape of the cells is such that two opposing honeycomb layers nest into each other, with each facet of the closed ends being shared by opposing cells.

Individual cells do not, of course, show this geometrical perfection: in a regular comb, there are deviations of a few percent from the "perfect" hexagonal shape. In transition zones between the larger cells of drone comb and the smaller cells of worker comb, or when the bees encounter obstacles, the shapes are often distorted.

In 1965, László Fejes Tóth discovered that the trihedral pyramidal shape (which is composed of three rhombi) used by the honey bee is not the theoretically optimal three-dimensional geometry. A cell end composed of two hexagons and two smaller rhombuses would actually be .035% (or approximately 1 part per 2850) more efficient. This difference is too minute to measure on an actual honeycomb, and irrelevant to the hive economy in terms of efficient use of wax, considering that wild comb varies considerably from any human notion of "ideal" geometry.

MARC FORNES | THEVERYMANY

MARC FORNES (registered Architect DPLG) is the founder and principal of THEVERYMANY - www.theverymany.net - a design studio and collaborative research forum engaging the field of architecture via what he qualifies as “Explicit and Encoded protocols”.

In 2004 he graduated with a Master of Architecture and Urbanism from the Design Research Lab of the Architectural Association in London after having previously studied in France and Sweden (KTH).

Marc’s professional work experience in La Reunion, France, UK and the US includes SOM, Ross Lovegrove and Zaha Hadid Architects, where he was the project architect, from competition to tender documentation, for an experimental Mediatheque in Pau. During his three years on this project he directed the material research and geometrical development for the largest self-supported carbon fibre shell to date.

Marc – together with Francois Roche (R&Sie(n)) - is co-teaching “(n)certainties” for the 5th time – a graduate studio currently hosted at Columbia University (NYC) and also at the University of Southern California (Los Angeles); in Fall 2008 it was hosted at Die Angewandte (Vienna) as its Cross Over studio. Marc was invited professor at University of Michigan (spring 08) and has previously led many workshops and appeared as a guest critic at the Architectural Association, The Royal College of Art, Columbia University, Pratt Institute, the University of Pennsylvania, Ball State University, in Chile, etc…

In parallel of academia - he is also teaching professional Rhinoscript classes for McNeel US and Europe.

In 2007 he produced & curated Scriptedbypurpose – www.scriptedbypurpose.net – the first exhibition exclusively focusing on scripted processes within design – and recently curated the European section for the Architecture Biennale 2008 in Beijing.

THEVERYMANY research – mostly showcased as experimental installation work - has been exhibited in many shows and art galleries around the world including London, Berlin, Frankfurt, Calgary, Lyon, New York, Los Angeles, Beijing…

Marc and THEVERYMANY are now based in New York – practicing both in the US and Europe – as architect/designer focusing mainly on computation, especially regarding the generation and automaton of parts…